On the Existence of Hereditarily G-Permutable Subgroups in Exceptional Groups G of Lie Type

A subgroup A of a group G is G -permutable in G if for every subgroup B≤ G there exists x∈ G such that ABˣ=BˣA . A subgroup A is hereditarily G -permutable in G if A is E -permutable in every subgroup E of G which includes A . The Kourovka Notebook has Problem 17.112(b): Which finite nonabelian simple groups G possess a proper hereditarily G -permutable subgroup? We answer this question for the exceptional groups of Lie type. Moreover, for the Suzuki groups G²\!B₂(q) we prove that a proper subgroup of G is G -permutable if and only if the order of the subgroup is 2. In particular, we obtain an infinite series of groups with G -permutable subgroups.